On the learnability of vector spaces

Abstract: The central topic of the paper is the learnability of the recursively enumerable subspaces of , where is the standard recursive vector space over the rationals with (countably) infinite dimension and V is a given recursively enumerable subspace of . It is shown that certain types of vector spaces can be characterized in terms of learnability properties: is behaviourally correct learnable from text iff V is finite-dimensional, is behaviourally correct learnable from switching the type of information iff V is finite-dimensional, 0-thin or 1-thin. On the other hand, learnability from an informant does not correspond to similar algebraic properties of a given space. There are 0-thin spaces and such that is not explanatorily learnable from an informant, and the infinite product is not behaviourally correct learnable from an informant, while both and the infinite product are explanatorily learnable from an informant. [Copyright &y& Elsevier]